Multidegrees of tame automorphisms in dimension three (Q409064)

Given a polynomial automorphism \(F\) of \(K[X,Y,Z]\) where \(K\) is a field of characteristic zero, then the multidegree of \(F=(F_1,F_2,F_3)\) is the sequence \((\deg(F_1),\deg(F_2),\deg(F_3))\). If \(F\) is tame, then by results of Umirbaev-Shestakov and follow-up work by Karas, shows that some multidegrees cannot belong to a tame map (while there are automorphisms having such degrees).NEWLINENEWLINEIn this article, the authors discuss when a sequence of positive integers can be the multidegree of some tame automorphism in dimension three, and they also relate these investigations to the problem of whether there exists a tame automorphism admitting a reduction of type II or type III.